On contractible edges in 3-connected graphs

The existence of contractible edges is a very useful tool in graph theory. For 3-connected graphs with at least six vertices, Ota and Saito (1988) prove that the set of contractible edges cannot be covered by two vertices. Saito (1990) prove that if a three-element vertex set S covers all contractible edges of a 3-connected graph G, then S is a vertex-cut of G provided that G has at least eight vertices. Using Saito’s result, Hemminger and Yu (1993) characterize all 3-connected graphs having at least ten vertices which has a 3-element vertex set covering all contractible edges. We give a direct short proof of the last result. Saito’s result is a consequence. We also give a short proof of the main results given by Ando, Enomoto and Saito (1987) and McCuaig (1990).